Quantum error-correction failure distributions: Comparison of coherent and stochastic error models

We compare failure distributions of quantum error correction circuits for stochastic errors and coherent errors. We utilize a fully coherent simulation of a fault-tolerant quantum error correcting circuit for a $d=3$ Steane and surface code. We find that the output distributions are markedly different for the two error models, showing that no simple mapping between the two error models exists. Coherent errors create very broad and heavy-tailed failure distributions. This suggests that they are susceptible to outlier events and that mean statistics, such as pseudothreshold estimates, may not provide the key figure of merit. This provides further statistical insight into why coherent errors can be so harmful for quantum error correction. These output probability distributions may also provide a useful metric that can be utilized when optimizing quantum error correcting codes and decoding procedures for purely coherent errors.

Figure 1

Histograms of the Steane QEC failure metrics, from $3\times {10}^6$ trials of the Steane QEC circuit, using the Pauli error model with $p=0.00005,0.0001,0.0002,0.0005,0.001,0.002,0.005,0.01,0.02$. $P_{\mathrm{fail}}^{(L+1)}$ (a) is always binomially distributed, but $P_{\mathrm{fail}}^{(\psi +2)}$ (b) and $P_{\mathrm{fail}}^{(\psi +1)}$ (c) never are.

Figure 2

The Monte Carlo simulation of the failure rate for the Steane QEC circuit, using the Pauli error model. The red circles are the fraction of trials with $P_{\mathrm{fail}}^{(L+1)}=1$. The blue and green symbols mark estimates of failure, based on the criteria of $P_{\mathrm{fail}}^{(\psi +2)}>{10}^{-6}$ and $P_{\mathrm{fail}}^{(\psi +1)}>{10}^{-6}$. The vertical bars represent the 95% confidence level in the estimates. The black line marks the pseudothreshold criteria of $P_{\mathrm{fail}}\left(p\right)=p$.

Figure 3

Shown are five different runs of ten consecutive cycles of the Steane [[7,1,3]] QEC circuit, using the Pauli error model. (a) Plots of the three metrics for QEC failure under consideration. (b) Plots of ${\mathcal{F}}^2$ and $P_{\mathrm{code}}$.

Figure 4

Histograms of the Steane QEC circuit failure metrics, for $1\times {10}^6$ trials of $\sigma =0.001$, 0.002, 0.005, 0.010, 0.020, 0.050, and 0.100. The curves proceed from left to right in the bottom panel.

Figure 5

Using the criteria $P_{\mathrm{fail}}^{(\psi +1)}>{10}^{-6}$ on the ${10}^6$ trials of the Steane QEC circuit, under the pulse-area error model, gives rise to this failure rate curve. The failure rate does not follow the expected linear slope. The black line is the unit line.

Figure 6

The success metrics $\mathcal{F}, P_{\mathrm{code}}$, and the ratio, for the pulse-area error model, acting on the Steane [[7,1,3]] QEC circuit. Values of $\sigma$ correspond to those of Fig. 4.

Figure 7

The tilted-13 surface code (distance 3). A $3\times 3$ layout of data qubits and a set of weight-2 and weight-4 stabilizers that, with an appropriate circuit schedule, is a fault-tolerant implementation of an error correcting code.

Figure 8

Histograms of the failure criteria for the tilted surface code with the Pauli error model. $1\times {10}^6$ samples were accumulated per error rate. Pauli error rates: $p=0.000001$, 0.0000025, 0.000005, 0.0000075, 0.00001, 0.000025, 0.00005, 0.000075, 0.0001, 0.0002, 0.0004, 0.0006, and 0.0008 generate the curves from left to right, respectively.

Figure 9

Histograms of the failure criteria for the tilted surface code with the pulse-area error model. $1\times {10}^6$ samples were accumulated per error rate. Pulse area error strengths: $\sigma =0.0025$, 0.005, 0.0075, 0.01, 0.02, 0.03, 0.04, and 0.05 generate the curves from left to right.